Abstract

We prove the strong law of large numbers for weighted sums , which generalizes and improves the corresponding one for independent and identically distributed random variables and -mixing random variables. In addition, we present some results on complete convergence for weighted sums of -mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.

1. Introduction

Throughout the paper, we let be the indicator function of the set . We assume that is a positive increasing function on satisfying as and is the inverse function of . Since , it follows that . For easy notation, we let and let . denotes that there exists a positive constant such that . denotes a positive constant.

Let be a sequence of independent observations from a population distribution. A common expression for these linear statistics is , where the weights are either real constants or random variables independent of . Many authors have studied the strong convergence properties for linear statistics and obtained some interesting results. For the details, one can refer to Bai and Cheng [1], Sung [2], Cai [3], Jing and Liang [4], Zhou et al. [5], Wang et al. [6–8] and Wu and Chen [9], Tang [10], and so forth.

Recently, Cai [11] proved the following strong law of large numbers for weighted sums of independent and identically distributed random variables.

Theorem A. Let be a sequence of independent and identically distributed random variables with , , and . Assume that the inverse function of satisfies
Let be a triangular array of constants such that(i);(ii) for some .Then for any ,

Wang et al. [12] generalized the result of Theorem A for independent sequences to the case of -mixing sequences under conditions of (1), , and . Conditions (1) and in Theorem A look ugly, since the conclusion (2) does not contain any information on . The main purpose of the paper is to show (2) for -mixing random variables without conditions of (1) and . So our result generalizes and improves the corresponding one of Cai [11] and Wang et al. [12]. In addition, we will present some results on complete convergence for weighted sums of -mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.

Firstly, let us recall the concept of -mixing random variables.

Let be a sequence of random variables defined on a fixed probability space . Write . Given -algebras in , let
Define the -mixing coefficients by
Obviously, , and .

Definition 1. A sequence of random variables is said to be -mixing if there exists such that .It is easily seen that -mixing (i.e., -mixing) sequence contains independent sequence as a special case. -mixing random variables were introduced by Bradley [13] and many applications have been found. -mixing is similar to -mixing, but both are quite different. Many authors have studied this concept providing interesting results and applications. See, for example, Bradley [13] for the central limit theorem, Bryc and Smoleński [14], Peligrad and Gut [15], and Utev and Peligrad [16] for moment inequalities, Gan [17], Kuczmaszewska [18], Wu and Jiang [19] and Wang et al. [20, 21] for almost sure convergence, Peligrad and Gut [15], Cai [22], Kuczmaszewska [23], Zhu [24], An and Yuan [25], Wang et al. [26], and Sung [27] for complete convergence, Peligrad [28] for invariance principle, Wu and Jiang [29] for strong limit theorems for weighted product sums of -mixing sequences of random variables, Wu and Jiang [30] for Chover-type laws of the -iterated logarithm, Wu [31] for strong consistency of estimator in linear model, Wang et al. [32] for complete consistency of the estimator of nonparametric regression models, Wu et al. [33] and Guo and Zhu [34] for complete moment convergence, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired. So studying the limit behavior of -mixing random variables is of interest.The following concepts of slowly varying function and stochastic domination will be used in this work.

Definition 2. A real-valued function , positive and measurable on , is said to be slowly varying if
for each .

Definition 3. A sequence of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such that
for all and .

This work is organized as follows. Some important lemmas are presented in Section 2. Main results and their proofs are provided in Section 3.

2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper. The first one is the Rosenthal type maximal inequality for -mixing random variables, which was obtained by Utev and Peligrad [16].

Lemma 4 (cf. Utev and Peligrad [16]). For a positive integer and positive real numbers and , there exists a positive constant such that if is a sequence of random variables with , , and for every , then for all ,

The next one is a basic property for stochastic domination. For the details of the proof, one can refer to Wu [35] or Tang [36].

Lemma 5. Let be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following two statements hold:
where and are positive constants. Consequently, , where is a positive constant.

The last one is the basic properties for slowly varying function, which was obtained by Bai and Su [37].

Lemma 6 (cf. Bai and Su [37]). If is a slowly varying function as , then(i) for each ;(ii);(iii), for each ;(iv) for every , , positive integer and some , ;(v) for every , , positive integer and some , .

3. Main Results and Their Proofs

In this section, we will generalize and improve the result of Theorem A for independent and identically distributed random variables to the case of -mixing random variables. In addition, we will present some results on complete convergence for weighted sums of -mixing random variables.

Our main results are as follows.

Theorem 7. Let be a sequence of -mixing random variables, which is stochastically dominated by a random variable with , and . Let be a triangular array of constants such that
Then for any ,

Proof. By Markov’s inequality, Lemmas 4 and 5, and condition (9), we have
which implies (10). This completes the proof of the theorem.

Remark 8. The key to the proof of Theorem 7 is the Rosenthal type maximal inequality for -mixing sequences (i.e., Lemma 4). Similar to the proof of Theorem 7, we have the following result.

Theorem 9. Let be a sequence of random variables, which is stochastically dominated by a random variable with and . Let be a triangular array of constants such that for some . Suppose that there exists a positive constant such that
Then (10) holds for any .

If the array of constants is replaced by the sequence of constants , then we can get the following strong law of large numbers for weighted sums . The proof is standard, so we omit the details.

Theorem 10. Let be a sequence of random variables, which is stochastically dominated by a random variable with and . Let be a sequence of constants such that for some . Suppose that there exists a positive constant such that
Then for any ,
and in consequence, .

Remark 11. There are many sequences of random variables satisfying (12), such as negatively associated (NA, in short) sequence (see Shao [38]), negatively superadditive-dependent (NSD, in short) sequence (see Wang et al. [39]), asymptotically almost negatively associated (AANA, in short) sequence (see Yuan and An [40]), -mixing sequence (see Wang et al. [12]), and -mixing sequence (see Utev and Peligrad [16]). Comparing Theorems 7 and 9 with Theorem A, conditions (1) and in Theorem A can be removed. In addition, the condition “identical distribution” in Theorem A can be weakened by “stochastic domination.” Hence, the results of Theorem 7 and Theorem 9 generalize and improve the corresponding one of Theorem A.In the following, we will present some results on complete convergence for weighted sums of -mixing random variables. The main ideas are inspired by Kuczmaszewska [41]. The first one is a very general result of complete convergence for weighted sums of -mixing random variables, which can be applied to obtain other result’s of complete convergence, such as Baum-Katz type complete convergence and Hsu-Robbins type complete convergence.

Theorem 12. Let be a sequence of -mixing random variables and let be an array of real numbers. Let be an increasing sequence of positive integers and let be a sequence of positive real numbers. If for some , and for any , the following conditions are satisfied:
then

Proof. Let
Therefore
Using the -inequality and Jensen’s inequality, we can estimate in the following way:
The desired result (18) follows from (15), (16), (17), (20), (21), and Lemma 4 immediately. The proof is completed.

Corollary 13. Let be a sequence of -mixing random variables and let be an array of real numbers. Let be a slowly varying function as , , and . If for some and , the following conditions are satisfied for any :
then

Proof. Let and let . The desired result (25) follows from conditions (22)–(24) and Theorem 12 immediately. The proof is completed.

If for and in Corollary 13, then we can get the following result for -mixing random variables.

Corollary 14. Let be a sequence of mean zero -mixing random variables, which is stochastically dominated by a random variable and, let be a slowly varying function as . If for some , and ,
then for any ,

Proof. The proof is similar to that of Corollary 2.8 in Kuczmaszewska [41]. Take , , , and . In order to prove (27), it is enough to prove that under, the conditions of Corollary 14, conditions (22) and (24) hold.In fact, by Lemma 6 and similar to the proof of Corollary 2.8 in Kuczmaszewska [41], we can obtain
which implies that (22) holds.Let be the distribution function of . It follows, by Lemmas 5 and 6 and the inequality above, that
where the fourth inequality above is followed by the proof of Corollary 2.8 in Kuczmaszewska [41]. This shows that (23) holds.By Lemma 5, -inequality, and Markov’s inequality, we can see that
where the last inequality is followed by the proof of Corollary 2.8 in Kuczmaszewska [41]. Hence, condition (24) is satisfied.The proof will be completed if we show that
If , then we have by Lemma 5 that
If , then it follows, by and Lemma 5, that
This completes the proof of the theorem.

Remark 15. Noting that, for typical slowly varying functions and , we can get the simpler formulas in the above theorems.

Corollary 16. Let and let be a sequence of -mixing random variables with and for . Let be an array of real numbers satisfying the condition
for some and . Then for any and ,

Proof. Take , , and in Theorem 12. Similar to the proof of Corollary 2.2 in Kuczmaszewska [41], we can see that conditions (15)–(17) in Theorem 12 are satisfied by (34).Noting that and , it follows, by for and (34), that
The desired result (35) follows from the statements above and Theorem 12 immediately. The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is most grateful to the Editor Ciprian A. Tudor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, and 1208085QA03), the 211 Project of Anhui University, the Youth Science Research Fund of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), the Students Science Research Training Program of Anhui University (kyxl2013003 and KYXL2012007), and the Students Innovative Training Project of Anhui University (201310357004).